Finite Group Databasebeta

← #21456817

Subgroups — #21456817

Order 7808.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16840317C211yesmaximal, characteristic, complemented
2#124×D61C211yesmaximal, characteristic
3#16840154C211yesmaximal, characteristic, complemented
4#16840122C211yesmaximal, characteristic, complemented
5#16840121C211yesmaximal, characteristic, complemented
6#16839986C212yesmaximal, complemented
7#16839982C212yesmaximal, complemented
8#16839916C212yesmaximal, complemented
9#16839911C212yesmaximal
10#16839817C211yesmaximal, characteristic, complemented
11C61×#497C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2211yescharacteristic
13C2×D4×D61C2211yescharacteristic
14C2×#11758202C2212yescomplemented
15C2×#11758201C2212yes—
16#14519777C2211yescharacteristic, complemented
17#14519768—21no—
18#14519766C2211yescharacteristic
19#14519765C2211yescharacteristic
20#14519764C2211yescharacteristic
21#14519763—21no—
22#14519763—21no—
23#14519757—21no—
24#14519753C2212yes—
25Q8×Dic61C2212yescomplemented
26#14519742—21no—
27#14519735—21no—
28C22⋊C4×D61—21no—
29C22⋊C4×D61—21no—
30C42×D61C2211yescharacteristic
31#14519706C2212yescomplemented
32#14519702C2212yescomplemented
33#14519701C2212yescomplemented
34#14519699C2212yes—
35#14519698C2212yes—
36#14519689C2212yes—
37#14519688C2211yescharacteristic, complemented
38C2×Q16×C61C2212yescomplemented
39C2×SD16×C61C2212yes—
40C4⋊Q8×C61C2211yescharacteristic, complemented
41#124×C61C2211yescharacteristic
42#97×C61C2211yescharacteristic, complemented
43C23×D61—21no—
44C2×Q8×C61C2311yescharacteristic
45C2×Q8×C61C2312yes—
46C2×D4×C61C2311yescharacteristic
47C2×C22⋊D61—21no—
48Q8×D61—42no—
49D4×D61—42no—
50C2×D244C2311yescharacteristic
51C2×C4×D61—21no—
52C2×C4×D61D411yescharacteristic
53C2×C4×D61D411yescharacteristic
54C2×C4×D61C2311yescharacteristic
55C2×Dic122—21no—
56C2×Dic122—21no—
57C2×Dic122C2311yescharacteristic
58Q16×C61—42no—
59SD16×C61—42no—
60C2×C8×C61C2312yes—
61C4⋊C4×C61—21no—
62C4⋊C4×C61—21no—
63C22⋊C4×C61—21no—
64C22⋊C4×C61—21no—
65C42×C61C2311yescharacteristic
66C22⋊Dic61—21no—
67C22⋊Dic61—21no—
68#11758202—42no—
69#11758201—42no—
70#11758198—22no—
71#11758198—21no—
72#11758198—21no—
73#11758198—21no—
74#11758198—21no—
75C4⋊Dic61—21no—
76C4⋊Dic61—22no—
77C4⋊Dic61C2312yes—
78#11758196—22no—
79C4×Dic61—22no—
80C4×Dic61D411yescharacteristic
81C4×Dic61D411yescharacteristic
82C4×Dic61C2311yescharacteristic
83C2×#92418C2312yes—
84C23×C61—21no—
85C22×D61—21no—
86C22×D61—42no—
87C22×D61—42no—
88C22×D61—42no—
89C22×D61C2×D411yescharacteristic
90Q8×C61—22no—
91Q8×C61—22no—
92Q8×C61—22no—
93D4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—21no—
96C2×C4×C61C2×D411yescharacteristic
97C2×C4×C61C2×D411yescharacteristic
98C2×C4×C61C2411yescharacteristic
99C22⋊D61—42no—
100C2×Dic61—22no—
101C2×Dic61—21no—
102C2×Dic61C2×D411yescharacteristic
103C2×Dic61C2×D411yescharacteristic
104C2×Dic61C2×D411yescharacteristic
105D244—22no—
106C4×D61—22no—
107C4×D61—22no—
108C4×D61—41no—
109C4×D61—42no—
110Dic122—22no—
111Dic122—42no—
112Dic122—42no—
113C8×C61—22no—
114#92418—22no—
115C22×C61—41no—
116C22×C61—41no—
117C22×C61—41no—
118C22×C61C22×D411yescharacteristic
119C2×D61—21no—
120C2×D61—21no—
121C2×D61—41no—
122C2×D61—41no—
123C2×D61—41no—
124C2×D61—84no—
125C2×D61#13712yes—
126C4×C61—21no—
127C4×C61—21no—
128C4×C61—41no—
129C4×C61—42no—
130C4×C61H(5,𝔽2)12yes—
131Dic61—21no—
132Dic61—21no—
133Dic61—21no—
134Dic61—41no—
135Dic61—42no—
136#3215—611nomaximal, Sylow
137C2×C61—81no—
138C2×C61#53511yescharacteristic
139C2×C61C2×#13711yescharacteristic
140C2×C61#57511yescharacteristic
141D61—22no—
142D61—81no—
143#533—611no—
144C2×#124—611no—
145#497—611no—
146#497D6111yescharacteristic, complemented
147#489—611no—
148#489—611no—
149#464—612no—
150#464—612no—
151#451—612no—
152#450—612no—
153#432—611no—
154C61#321511yescharacteristic, complemented, Sylow
155C22×Q8—611no—
156C22×D4—611no—
157C2×Q16—612no—
158C2×Q16C2×D6112yescomplemented
159C2×SD16—612no—
160C2×SD16C2×D6112yes—
161C4⋊Q8—611no—
162C4⋊Q8C2×D6111yescharacteristic, complemented
163#125—1221no—
164#124—611no—
165#124—611no—
166#124—611no—
167#124—1221no—
168#124—1221no—
169#124C2×D6111yescharacteristic
170C22⋊Q8—612no—
171C22⋊Q8—1221no—
172C4×Q8—612no—
173C4×Q8—1221no—
174#117—1221no—
175C2×C22⋊C4—1221no—
176C2×C22⋊C4—1221no—
177C2×C42—611no—
178C4⋊C8—612no—
179#103—612no—
180#103—612no—
181#102—612no—
182#102—612no—
183C22⋊C8—612no—
184#97—611no—
185#97C2×D6111yescharacteristic, complemented
186C24—1221no—
187C2×Q8—611no—
188C2×Q8—1221no—
189C2×Q8—1221no—
190C2×Q8—2442no—
191C2×Q8C22×D6111yescharacteristic
192C2×Q8C22×D6112yes—
193C2×D4—611no—
194C2×D4—1221no—
195C2×D4—2442no—
196C2×D4C22×D6111yescharacteristic
197C22×C4—611no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—1221no—
201Q16—42no—
202Q16—2442no—
203SD16—42no—
204SD16—2442no—
205C2×C8—612no—
206C2×C8C22×D6112yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—612no—
210C4⋊C4—1222no—
211C4⋊C4—1222no—
212C4⋊C4—1221no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1221no—
216C22⋊C4—1221no—
217C22⋊C4—1222no—
218C22⋊C4—1221no—
219C22⋊C4—1221no—
220C22⋊C4—1221no—
221C22⋊C4—1221no—
222C42—611no—
223C42—611no—
224C42—611no—
225C42—1222no—
226C42C22×D6111yescharacteristic
227C23—21no—
228C23—611no—
229C23—1221no—
230C23—2442no—
231C23—2442no—
232C23—2442no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—1222no—
237Q8—2442no—
238Q8—2442no—
239D4—22no—
240D4—1222no—
241D4—2442no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—611no—
245C2×C4—611no—
246C2×C4—611no—
247C2×C4—1222no—
248C2×C4—1222no—
249C2×C4—1222no—
250C2×C4—1221no—
251C2×C4—2441no—
252C2×C4—2442no—
253C2×C4D4×D6111yescharacteristic
254C2×C4D4×D6111yescharacteristic
255C2×C4C23×D6111yescharacteristic
256C8—22no—
257C8—1222no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—612no—
262C22—1221no—
263C22—1221no—
264C22—4884no—
265C22—2441no—
266C22—2441no—
267C22—2441no—
268C22C2×D4×D6111yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—1221no—
274C4—1221no—
275C4—1221no—
276C4—2442no—
277C4—2441no—
278C4#1451982212yes—
279C2—81no—
280C2—1222no—
281C2—4881no—
282C2#1684032511yescharacteristic
283C2#137×D6111yescharacteristic
284C2#1684056111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#497C2
socleC22×C61C22×D4
hypercenter#497D61
hypocenterC61#3215
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16840317C2semidirect
3. #16840154C2semidirect
4. #16840122C2semidirect
5. #16840121C2semidirect
6. #16839986C2semidirect
7. #16839982C2semidirect
8. #16839916C2semidirect
10. #16839817C2semidirect
11. C61×#497C2semidirect
14. C2×#11758202C22semidirect
16. #14519777C22semidirect
25. Q8×Dic61C22semidirect
31. #14519706C22semidirect
32. #14519702C22semidirect
33. #14519701C22semidirect
37. #14519688C22semidirect
38. C2×Q16×C61C22semidirect
40. C4⋊Q8×C61C22semidirect
42. #97×C61C22semidirect
146. #497D61semidirect
154. C61#3215semidirect
158. C2×Q16C2×D61semidirect
162. C4⋊Q8C2×D61semidirect
185. #97C2×D61semidirect